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M. Wallner:
"Lattice paths below a line of rational slope";
Vortrag: MADACA - Random Walks and Dunkl Processes: Algebraic and Combinatorical Approaches, Domaine de Chalès, Frankreich (eingeladen); 20.06.2016 - 24.06.2016.

We analyze some enumerative and asymptotic properties of lattice paths below a line of rational slope. We illustrate our approach with Dyck paths under a line of slope 2/5. This answers Knuth's problem
#4 from his Flajolet lecture during the conference Analysis of Algorithms (AofA'2014) in Paris in June 2014. Our approach extends the work of Banderier and Flajolet for asymptotics and enumeration of directed lattice paths to the case of generating functions involving several dominant singularities. A key ingredient in the proof is the generalization of an old trick by Knuth himself (for enumerating permutations sortable by a stack), promoted by Flajolet and others as the kernel method. All the corresponding generating functions are algebraic, and they offer some new combinatorial identities, which can also be tackled in the A=B spirit of Wilf-Zeilberger-Petkovsek. We show how to obtain similar results for other slopes than 2/5. An interesting case being e.g. Dyck paths below the slope 2/3, which corresponds to the so called Duchon's club model. We also show how to use our techniques for lattice paths below an irrational slope and illustrate this with Dyck paths below y = x/sqrt(2).